(4) $$ \frac{2}{3}+\frac{3 x+2}{x}-\frac{x+4}{2 x} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2}{3} + \frac{3x + 2}{x} - \frac{x + 4}{2x}
$$

**Step 1: Find a Common Denominator**

The denominators are $$3$$, $$x$$, and $$2x$$. The least common denominator (LCD) for these is $$6x$$.

**Step 2: Rewrite Each Fraction with the LCD**

1. $$\frac{2}{3}$$:
   $$
   \frac{2}{3} = \frac{2 \times 2x}{3 \times 2x} = \frac{4x}{6x}
   $$

2. $$\frac{3x + 2}{x}$$:
   $$
   \frac{3x + 2}{x} = \frac{(3x + 2) \times 6}{x \times 6} = \frac{18x + 12}{6x}
   $$

3. $$\frac{x + 4}{2x}$$:
   $$
   \frac{x + 4}{2x} = \frac{(x + 4) \times 3}{2x \times 3} = \frac{3x + 12}{6x}
   $$

**Step 3: Combine the Fractions**

$$
\frac{4x}{6x} + \frac{18x + 12}{6x} - \frac{3x + 12}{6x} = \frac{4x + 18x + 12 - 3x - 12}{6x}
$$

**Step 4: Simplify the Numerator**

$$
4x + 18x + 12 - 3x - 12 = 19x
$$

**Step 5: Final Simplification**

$$
\frac{19x}{6x} = \frac{19}{6}
$$

**Conclusion:**

The simplified form of the expression is:

$$
\frac{19}{6}
$$

Answer: The expression simplifies to 19⁄6.
The expression simplifies to $$ \frac{19}{6} $$.

(4) \( \frac{2}{3}+\frac{3 x+2}{x}-\frac{x+4}{2 x} \)